Activity 2.8 Vertical Asymptotes

Whenever \(\lim\limits_{x \to a}\fe{f}{x} \neq 0\) but \(\lim\limits_{x \to a}\fe{g}{x} = 0\text{,}\) then \(\lim\limits_{x \to a}\frac{\fe{f}{x}}{\fe{g}{x}}\) does not exist because from either side of \(a\) the value of \(\frac{\fe{f}{x}}{\fe{g}{x}}\) has an absolute value that will become arbitrarily large. In these situations the line \(x = a\) is a vertical asymptote for the graph of \(y = \frac{\fe{f}{x}}{\fe{g}{x}}\text{.}\) For example, the line \(x = 2\) is a vertical asymptote for the function \(h\) defined by \(\fe{h}{x} = \frac{x+5}{2-x}\text{.}\) We say that \(\lim\limits_{x \to 2}\frac{x+5}{2-x}\) has the form “not-zero over zero.” (Specifically, the form of \(\lim\limits_{x \to 2}\frac{x+5}{2-x}\) is \(\frac{7}{0}\text{.}\)) Every limit with form “not-zero over zero” does not exist. However, we frequently can communicate the non-existence of the limit using an infinity symbol. In the case of \(\fe{h}{x} = \frac{x+5}{2-x}\) it’s pretty easy to see that \(\fe{h}{1.99}\) is a positive number whereas \(\fe{h}{2.01}\) is a negative number. Consequently, we can infer that \(\lim\limits_{x \to 2^{-}}\fe{h}{x} = \infty\) and \(\lim\limits_{x \to 2^{+}}\fe{h}{x} = -\infty\text{.}\) Remember, these equations are communicating that the limits do not exist as well as the reason for their non-existence. There is no short-hand way to communicate the non-existence of the two-sided limit \(\lim\limits_{x \to 2}\fe{h}{x}\text{.}\)